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Properties

Label 16T42
Degree $16$
Order $32$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_4\wr C_2$

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Show commands: Magma

magma: G := TransitiveGroup(16, 42);
 

Group action invariants

Degree $n$:  $16$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $42$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_4\wr C_2$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $8$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,3,5,15)(2,4,6,16)(7,8)(9,10)(11,12)(13,14), (1,10)(2,9)(3,12)(4,11)(5,14)(6,13)(7,16)(8,15)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $D_{4}$ x 2, $C_4\times C_2$
$16$:  $C_2^2:C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$, $D_{4}$ x 2

Degree 8: $D_4$, $C_4\wr C_2$ x 2

Low degree siblings

8T17 x 2, 16T28, 32T14

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderRepresentative
$1^{16}$ $1$ $1$ $()$
$2^{4},1^{8}$ $2$ $2$ $( 7,11)( 8,12)( 9,13)(10,14)$
$4^{2},2^{4}$ $2$ $4$ $( 1, 2)( 3, 4)( 5, 6)( 7, 9,11,13)( 8,10,12,14)(15,16)$
$4^{2},2^{4}$ $2$ $4$ $( 1, 2)( 3, 4)( 5, 6)( 7,13,11, 9)( 8,14,12,10)(15,16)$
$4^{2},2^{4}$ $2$ $4$ $( 1, 3, 5,15)( 2, 4, 6,16)( 7,12)( 8,11)( 9,14)(10,13)$
$4^{4}$ $1$ $4$ $( 1, 4, 5,16)( 2, 3, 6,15)( 7,10,11,14)( 8, 9,12,13)$
$4^{4}$ $2$ $4$ $( 1, 4, 5,16)( 2, 3, 6,15)( 7,14,11,10)( 8,13,12, 9)$
$2^{8}$ $1$ $2$ $( 1, 5)( 2, 6)( 3,15)( 4,16)( 7,11)( 8,12)( 9,13)(10,14)$
$4^{2},2^{4}$ $2$ $4$ $( 1, 6)( 2, 5)( 3,16)( 4,15)( 7,13,11, 9)( 8,14,12,10)$
$2^{8}$ $4$ $2$ $( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)(13,15)(14,16)$
$4^{4}$ $4$ $4$ $( 1, 7, 5,11)( 2, 8, 6,12)( 3, 9,15,13)( 4,10,16,14)$
$8^{2}$ $4$ $8$ $( 1, 8, 4, 9, 5,12,16,13)( 2, 7, 3,10, 6,11,15,14)$
$8^{2}$ $4$ $8$ $( 1, 8,16,13, 5,12, 4, 9)( 2, 7,15,14, 6,11, 3,10)$
$4^{4}$ $1$ $4$ $( 1,16, 5, 4)( 2,15, 6, 3)( 7,14,11,10)( 8,13,12, 9)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $32=2^{5}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:  $3$
Label:  32.11
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 4A1 4A-1 4B 4C1 4C-1 4D1 4D-1 4E 8A1 8A-1
Size 1 1 2 4 1 1 2 2 2 2 2 4 4 4
2 P 1A 1A 1A 1A 2A 2A 2B 2B 2A 2B 2B 2A 4A1 4A-1
Type
32.11.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
32.11.1b R 1 1 1 −1 1 1 1 −1 −1 −1 −1 −1 1 1
32.11.1c R 1 1 1 −1 1 1 1 1 1 1 1 −1 −1 −1
32.11.1d R 1 1 1 1 1 1 1 −1 −1 −1 −1 1 −1 −1
32.11.1e1 C 1 1 −1 −1 −1 −1 1 −i i i −i 1 −i i
32.11.1e2 C 1 1 −1 −1 −1 −1 1 i −i −i i 1 i −i
32.11.1f1 C 1 1 −1 1 −1 −1 1 −i i i −i −1 i −i
32.11.1f2 C 1 1 −1 1 −1 −1 1 i −i −i i −1 −i i
32.11.2a R 2 2 −2 0 2 2 −2 0 0 0 0 0 0 0
32.11.2b R 2 2 2 0 −2 −2 −2 0 0 0 0 0 0 0
32.11.2c1 C 2 −2 0 0 −2i 2i 0 −1−i −1+i 1−i 1+i 0 0 0
32.11.2c2 C 2 −2 0 0 2i −2i 0 −1+i −1−i 1+i 1−i 0 0 0
32.11.2d1 C 2 −2 0 0 −2i 2i 0 1+i 1−i −1+i −1−i 0 0 0
32.11.2d2 C 2 −2 0 0 2i −2i 0 1−i 1+i −1−i −1+i 0 0 0

magma: CharacterTable(G);